In terms of cocycle models, classes of KR-theory are represented by complex vector bundles over XX on which the involution on their base space lifts to an anti-linear involution of the total space. Over manifolds with trivial involution these are precisely the complexification of real vector bundles and hence over such spaces KRKR-theory reduces to KO-theory. Conversely, over two copies X∪XX \cup X lof XX equipped with the involution that interchanges the two, KRKR-theory reduces to KU-theory. Finally over X×S1X \times S^1 with the involution the antipodal identification on the second (circle) factor , KRKR-theory reduces to the self-conjugate KSC-theory (Anderson 64). So in general KRKR-theory interpolates between all these cases. For instance on X×S1X \times S^1 with the reflection-involution on the circle (the real space denoted S1,1S^{1,1}, the non-trivial ℤ2\mathbb{Z}_2-representation sphere) it behaves like KOKO-theory at the two involution fixed points (the two O-planes) and like KUKU in their complement (a model that makes this very explicit is given in DMR 13, section 4), schematically:

Remark on terminology

An involution on a space by a homeomorphism (or diffeomorphism) as they appear in KR theory may be thought of as a “non-linear real structure”, and therefore spaces equipped with such involutions are called “real spaces”. Following this, KRKR-theory is usually pronounced “real K-theory”. But beware that this terminology easily conflicts with or is confused with KO-theory. For disambiguation the latter might better be called “orthogonal K-theory”. But on abstract grounds maybe KRKR-theory would best be just called ℤ2\mathbb{Z}_2-equivariant complex K-theory.

Definition

As the Grothendieck group of complex vector bundles with real structure

Properties

Bigrading

As any genuine equivariant cohomology theoryKRKR-theory is naturally graded over the representation ringRO(ℤ2)RO(\mathbb{Z}_2). Write ℝ\mathbb{R} for the trivial 1-dimensional representation and ℝ−\mathbb{R}_- for that given by the sign involution. Then the general orthogonalrepresentation decomposes as a direct sum